Let the equation x(x+2)(12−k)=2 have equal roots. Then the distance of the point (k,2k) from the line 3x+4y+5=0 is:
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Step-by-Step Solution
Step 1: Rearrange the equation into standard quadratic form
First, we expand the given equation x(x+2)(12−k)=2 to transform it into the standard quadratic form ax2+bx+c=0. This involves multiplying the terms and moving the constant to the left side of the equation.
Step 2: Apply the condition for equal roots
For a quadratic equation to have equal roots, its discriminant must be zero. The discriminant is given by the formula D=b2−4ac. We substitute the coefficients a=(12−k), b=2(12−k), and c=−2 into this formula and set it to zero.
Step 3: Solve for k
We simplify the equation obtained from the discriminant condition. Factoring out 4(12−k) allows us to easily find the possible values for k. This gives us two potential values for k: 12 and 14. However, if k=12, the original equation becomes −2=0, which is not possible. Therefore, k=14 is the only valid solution.
Step 4: Determine the coordinates of the point
Now that we have found the value of k=14, we can determine the exact coordinates of the given point (k,2k) by substituting k=14 into the expression.
Step 5: Calculate the distance from the point to the line
Finally, we use the formula for the perpendicular distance from a point (x1,y1) to a line Ax+By+C=0. We substitute the point (14,7) and the line 3x+4y+5=0 into the formula to calculate the distance.