<p>We study the proportion of conics given by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\mathcal {C}_{\varvec{F}, \varvec{y}}): F_0(\varvec{y})x_0^2 + F_1(\varvec{y})x_1^2 = F_2( \varvec{y})x_2^2 \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">C</mi> <mrow> <mrow> <mi mathvariant="bold-italic">F</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>x</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> which have a rational point <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{x} = (x_0:x_1:x_2) \in \mathbb {P}^2(\mathbb {Q})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>:</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>:</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{y} = (y_0: \dots : y_n)\in \mathbb {P}^n(\mathbb {Q})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>:</mo> <mo>⋯</mo> <mo>:</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(F_0,F_1,F_2 \in \mathbb {Z}[X_0,\ldots , X_n]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are homogeneous polynomials in many variables of the same degree <i>d</i>. We provide an asymptotic formula for the number of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varvec{y}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> </math></EquationSource> </InlineEquation> of bounded height such that the corresponding conic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((\mathcal {C}_{\varvec{F}, \varvec{y}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">C</mi> <mrow> <mrow> <mi mathvariant="bold-italic">F</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">y</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has a rational point. In particular, our result agrees with the Loughran–Smeets and the Loughran–Rome–Sofos conjectures. Our strategy is based on a recent result of Destagnol–Lyczak–Sofos relying on the circle method to estimate the average of an arithmetic function over polynomials in many variables. To this end, we study the proportion of conics <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(t_0x_0^2 + t_1x_1^2 + t_2x_2^2 = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <msubsup> <mi>x</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> having a rational point, and coefficients <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(t_0,t_1,t_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> in arithmetic progressions.</p>