If two tangents drawn from a point (α, β) lying on the ellipse 25x² + 4y² = 1 to the parabola y² = 4x are such that the slope of one tangent is four times the other, then the value of (10α + 5)² + (16β² + 50)² equals ___________.

Solution

Maths sir · 5 steps

Write the tangent as y = mx + 1/m and let the point give a quadratic in slope m.

1. Slope form. For y² = 4x we have a = 1, so every tangent is y = mx + 1/m. It passes through (α, β), so β = mα + 1/m, which gives α m² - β m + 1 = 0.
2. Roots m and 4m. The two slopes are the roots, so let them be m and 4m. Then the sum is 5m = β/α and the product is 4m² = 1/α.
3. Eliminate m. Squaring the sum gives 25m² = β²/α². Put in m² = 1/4α to get 25/4α = β²/α², so β² = 25α/4.
4. Use the ellipse. The point lies on 25x² + 4y² = 1, so 25α² + 4β² = 1. With 4β² = 25α this becomes 25α² + 25α = 1. There is no need to solve for α.

   (10α+5)² = 4(25α²+25α) + 25 = 29; (16β²+50)² = (100α+50)² = 400(25α²+25α) + 2500 = 2900

5. So. The required value is 29 + 2900 = 2929.

The idea to keep. Tangents from a point to a parabola: put the point into the slope form and get a quadratic in m. Any condition on the slopes then becomes a relation through the sum and product of the roots.

Answer from the NTA final answer key. Solution by PrepWiser.