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Maths Question 4 – JEE-MAIN 2025

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Let the equation x(x+2)(12−k)=2 have equal roots. Then the distance of the point (k,k2) from the line 3x+4y+5=0 is

For a quadratic equation to have equal roots, its discriminant must be zero.

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Ninja Strategy•Simplify Distance Formula First

Simplify the distance formula for the given point and line to d=|k+1|. This immediately shows the distance must be an integer, eliminating irrational options. Then, solve for k from the quadratic condition and substitute.

Step 1: Formulate the Quadratic Equation and Apply Equal Roots Condition✦ Active

The given equation is x(x+2)(12−k)=2. Expand and rearrange it into the standard quadratic form Ax2+Bx+C=0:

(x2+2x)(12−k)=2 (12−k)x2+2(12−k)x−2=0

For equal roots, the discriminant D=B2−4AC must be zero. Here, A=(12−k), B=2(12−k), and C=−2. Let m=(12−k).

(2m)2−4(m)(−2)=0 4m2+8m=0 4m(m+2)=0

This yields m=0 or m=−2. If m=0, then 12−k=0⟹k=12. In this case, the original equation becomes 0=2, which is impossible, so m≠0. Therefore, m=−2.

12−k=−2⟹k=14
Step 2: Determine the Point Coordinates○ Expand

The point is given as (k,k2). Substitute the value of k=14 into the coordinates:

(14,142)=(14,7)
Step 3: Calculate the Distance from the Point to the Line○ Expand

The line equation is 3x+4y+5=0. The distance d from a point (x1,y1) to a line Ax+By+C=0 is given by the formula d=|Ax1+By1+C|A2+B2. Here, (x1,y1)=(14,7), A=3, B=4, C=5.

d=|3(14)+4(7)+5|32+42 d=|42+28+5|9+16 d=|75|25 d=755 d=15
💡 Teacher's Secret Hint

Remember to take the absolute value of the numerator in the distance formula.

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